Questions and Answers
A linear transformation is a special type of function.
True
If A is a 3x5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is R3.
False
If A is an mxn matrix, then the range of the transformation x-->Ax is Rm.
False
Every linear transformation is a matrix transformation.
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A transformation T is linear if and only if T(c1v1+c2v2)=c1T(v1)+c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.
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A linear transformation T:Rn-->Rm is completely determined by its effect on the columns of the nxn identity matrix.
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If T:R2-->R2 rotates vectors about the origin through an angle x then T is a linear transformation.
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When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
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A mapping T: Rn-->Rm is onto if every vector x in Rn maps onto some vector in Rm.
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If A is a 3x2 matrix, then the transformation x-->Ax cannot be one-to-one.
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What can you say about the columns of A if the last column of AB is entirely zero but B itself has no columns of zero?
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If A and B are 2x2 with columns a1, a2 and b1, b2, respectively, then AB=[a1b1 a2b2].
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Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
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AB+AC=A(B+C).
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A^T+B^T=(A+B)^T.
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The transpose of a product of matrices equals the product of their transposes in the same order.
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In order for a matrix B to be the inverse of A, both equations AB=I and BA=I must be true.
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If A and B are nxn and invertible, then A^-1*B^-1 is the inverse of AB.
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If the determinant equals zero, then A is invertible.
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If A is an invertible nxn matrix, then the equation Ax=B is consistent for each B in Rn.
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Each elementary matrix is invertible.
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If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix.
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If the columns of A span Rn, then the columns are linearly independent.
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If A is an nxn matrix, then the equation Ax=b has at least one solution for each b in Rn.
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If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
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If A^T is not invertible, then A is not invertible.
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Can a square matrix with two identical columns be invertible? Why or why not?
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If A is invertible then the columns of A^-1 are linearly independent. Explain why.
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If a nxn matrix K cannot be row reduced to the identity matrix, what can you say about the columns of K? Why?
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An nxn determinant is defined by determinants of (n-1)x(n-1) submatrices.
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The (i,j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its ith row and jth column.
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A row replacement operation does not affect the determinant of a matrix.
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The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U.
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If the columns of A are linearly dependent, the detA=0.
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Det(A+B)=detA+detB.
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A vector is any element of a vector space.
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If u is a vector in a vector space V, the (-1)u is the same as the negative of u.
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A vector space is also a subspace of itself.
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R2 is a subspace of R3.
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A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u+v are in H, and (iii) c is a scalar and cu is in H.
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A null space is a vector space.
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The column space of an mxn matrix is in Rm.
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The column space of A, Col(A), is the set of all solutions of Ax=b.
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The null space of A, Nul(A), is the kernel of the mapping x-->Ax.
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The range of a linear transformation is a vector space.
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The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
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Every elementary row operation is reversible.
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A 5x6 matrix has six rows.
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The solution set of a linear system involving variables x1,...,xn is a list of numbers (s1,...,sn) that makes each equation in the system a true statement when the values of s1,...,sn are substituted for x1,...,xn respectively.
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Two fundamental questions about a linear system involve existence and uniqueness.
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In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
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The row reduction algorithm applies only to augmented matrices for a linear system.
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A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
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Finding a parametric description of the solution set of a linear system is the same as solving the system.
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If one row in an echelon form of an augmented matrix is inconsistent, then the associated linear system is inconsistent.
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Suppose a 6x8 coefficient matrix for a system has 6 pivot columns. Is the system consistent?
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Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
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A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution? Explain.
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A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. Can such a system be consistent?
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The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution.
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If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.
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The columns of any 4x5 matrix are linearly dependent.
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If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span{x,y}.
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A linearly independent set in a subspace H is a basis for H.
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If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.
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A basis is a linearly independent set that is as large as possible.
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The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.
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If B is an echelon form of a matrix A, then the pivot columns of B form a basis for ColA.
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The set R2 is a two dimensional subspace of R3.
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The number of variables in the equation Ax=0 equals the dimension of Nul A.
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A vector space is infinite-dimensional if it is spanned by an infinite set.
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If dimV=n and if S spans V, then S is a basis of V.
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The only three dimensional subspace of R3 is R3 itself.
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Study Notes
Linear Transformations
- A linear transformation maps vectors from Rn to Rm, defined by T(x) = Ax.
- It is not accurate to assert that all linear transformations are matrix transformations; matrix transformations are a specific case.
- The properties of linearity can be tested using the equation T(c1v1 + c2v2) = c1T(v1) + c2T(v2).
Matrix Operations and Properties
- A transformation T defined by a matrix A can only be one-to-one if no free variables exist in the equation Ax = b.
- The range of a transformation is defined by the linear combinations of the matrix's columns.
- The distributive law for matrices states that A(B+C) = AB + AC, and transposition properties include A^T + B^T = (A + B)^T.
Invertibility and Determinants
- A matrix A is invertible only if the determinant of A is non-zero; otherwise, it is not invertible.
- The relationship between a matrix and its transpose offers crucial insight into invertibility: if A^T is not invertible, then A is not invertible.
- An elementary matrix is always invertible, and each elementary row operation corresponds to an operation that can be reversed.
Linear Independence and Span
- A set of vectors is linearly independent if no vector can be expressed as a combination of others in the set.
- If the columns of a matrix A span Rn, they must also be linearly independent. Conversely, if they are linearly dependent, the determinant equals zero.
- The pivot positions in an echelon form of a matrix relate directly to the solution properties of systems involving that matrix.
Subspaces and Basis
- A subspace of a vector space must contain the zero vector and be closed under vector addition and scalar multiplication.
- For a collection of vectors to serve as a basis for a vector space, they must be linearly independent and span the space.
- The null space and column space are fundamental subspaces associated with matrices.
Systems of Equations
- The consistency of a linear system is often determined by the presence of pivot positions in the augmented matrix.
- A linear system with more equations than unknowns can be consistent but may not guarantee a unique solution.
- The nature of solutions to linear systems is connected to the concepts of basic and free variables, with basic variables corresponding to pivot positions.
General Theorems and Concepts
- The invertible matrix theorem encapsulates essential properties of invertibility, including implications for solution existence and uniqueness.
- Cohesions between linear transformations and their algebraic representations are vital for understanding dimensions and vector spaces.
- The properties of determinants, including those derived from row operations, are essential for solving and analyzing linear equations.
Vector Spaces
- A vector space is defined as infinite-dimensional if it can be spanned by an infinite set of vectors.
- A finite set of vectors that spans a vector space guarantees that some subset of that set constitutes a basis for the space.
- The columns of a matrix being linearly dependent arise from the matrix's inability to reduce to an identity matrix during row reduction processes.
Studying That Suits You
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Test your knowledge of linear transformations and matrices with these true or false flashcards. This quiz includes key concepts and definitions essential for mastering Linear Algebra. Challenge yourself and see how well you understand the fundamentals!
