Linear Algebra Midterm Review

If A is a 3X3 matrix, the volume of the parallelepiped determined by the columns of A is |det A|.

Linear combination refers to any sum of scalar multiples of vectors.

Span {v1,....,vp} denotes the set of all vectors that can be written as linear combinations of v1,....,vp.

<p>You take the RREF form of a matrix and any non-pivot position columns are part of the null space of the matrix.</p> Signup and view all the answers

<p>You take the RREF form of a matrix and any pivot position columns are part of the column space of the matrix.</p> Signup and view all the answers

<p>A linear transformation T from a vector space V into a vector space W assigns to each vector x in V a unique vector T(x) in W.</p> Signup and view all the answers

<p>An indexed set of vectors {v1,....,vp} in V is linearly independent if the equation c1v1+c2v2+....+cpvp=0 has only the trivial solution.</p> Signup and view all the answers

<p>The set {v1,....,vp} is linearly dependent if c1v1+c2v2+....+cpvp=0 has a nontrivial solution.</p> Signup and view all the answers

<p>A basis for H is an indexed set of vectors B={b1,....,bp} in V that is linearly independent and spans H.</p> Signup and view all the answers

<p>The coordinates of x relative to the basis B (or B-coordinates of x) are the weights c1,....,cn such that x=c1b1+...+cnbn.</p> Signup and view all the answers

<p>The coordinate mapping assumes that the basis B is an indexed set whose vectors are listed in a fixed order.</p> Signup and view all the answers

<p>PB is the change-of-coordinates matrix from B to the standard basis in Rn.</p> Signup and view all the answers

<p>A one-to-one linear transformation from a vector space V onto a vector space W is called an isomorphism.</p> Signup and view all the answers

<p>If two vector spaces are isomorphic, there is an isomorphism between them.</p> Signup and view all the answers

<p>Dimension of V, written as dim V, is the number of vectors in a basis for V.</p> Signup and view all the answers

<p>If V is spanned by a finite set, then V is said to be finite-dimensional.</p> Signup and view all the answers

<p>If V is not spanned by a finite set, then V is said to be infinite-dimensional.</p> Signup and view all the answers

<p>The trivial vector space contains only the zero vector, and its dimension is zero.</p> Signup and view all the answers

<p>A system of linear equations is a collection of one or more linear equations involving the same variables.</p> Signup and view all the answers

<p>A solution is a list of numbers that makes each equation a true statement when substituted for the variables.</p> Signup and view all the answers

<p>The solution set is the set of all possible solutions to the linear system.</p> Signup and view all the answers

<p>Two systems are equivalent if they have the same solution set.</p> Signup and view all the answers

<p>A system is consistent if it has either one solution or infinitely many solutions.</p> Signup and view all the answers

<p>A system is inconsistent if it has no solution.</p> Signup and view all the answers

<p>The coefficient matrix contains the coefficients of each variable aligned in the columns.</p> Signup and view all the answers

<p>The augmented matrix is formed by combining the coefficient matrix and the solution vector.</p> Signup and view all the answers

<p>The size of a matrix is denoted as an mXn matrix.</p> Signup and view all the answers

<ol> <li>Replace one row by the sum of itself and a multiple of another row. 2. Interchange two rows. 3. Multiply all entries in a row by a nonzero constant.</li> </ol> Signup and view all the answers

<p>Two matrices are row equivalent if one can be transformed into the other using elementary row operations.</p> Signup and view all the answers

<p>If the augmented matrices are row equivalent, then the two systems have the same solution set.</p> Signup and view all the answers

<p>Existence refers to having at least one solution, while uniqueness means having exactly one solution without free variables.</p> Signup and view all the answers

<p>There can be infinitely many solutions.</p> Signup and view all the answers

<p>In RREF, leading entries in nonzero rows are 1s, and each leading 1 is the only nonzero entry in its column.</p> Signup and view all the answers

<p>Get a leading 1 in the (1,1) entry, then zeros below and above it in the column, continuing down the diagonal.</p> Signup and view all the answers

<p>No, there will only be one RREF for any given matrix.</p> Signup and view all the answers

<p>Identify them by the pivot positions in the matrix.</p> Signup and view all the answers

<p>The solutions change in the same manner as the matrix by ERO.</p> Signup and view all the answers

<p>You obtain the RREF and augment the solution vector in the matrix.</p> Signup and view all the answers

<p>A pivot position is where there is a leading 1 in the reduced echelon form of the matrix.</p> Signup and view all the answers

<p>A pivot column is a column that contains a pivot position.</p> Signup and view all the answers

<p>They provide a basis for the column space of the matrix.</p> Signup and view all the answers

<p>R^n is the collection of all lists (or ordered n-tuples) of n real numbers.</p> Signup and view all the answers

<p>A linear combination is formed by c1v1 + ... + cpvp for given vectors and scalars.</p> Signup and view all the answers

<p>The span is the collection of all vectors that can be expressed as c1v1 + c2v2 + ... + cpvp.</p> Signup and view all the answers

<p>Take the coefficients of each variable and arrange them into a matrix for the equation.</p> Signup and view all the answers

<p>Ensure that the number of columns in the first matrix equals the number of rows in the second. Multiply corresponding entries row-wise.</p> Signup and view all the answers

<p>A(u + v) = Au + Av; A(cu) = c(Au).</p> Signup and view all the answers

<p>The number of columns in the first matrix must equal the number of rows in the second. Multiply and sum appropriately.</p> Signup and view all the answers

<p>Associative, left distributive, right distributive, and identity properties define matrix multiplication.</p> Signup and view all the answers

<p>A homogeneous system can be expressed as Ax = 0.</p> Signup and view all the answers

<p>A non-homogeneous system can be written as Ax ≠ 0.</p> Signup and view all the answers

<p>The zero solution is usually called the trivial solution.</p> Signup and view all the answers

<p>A nontrivial solution is a nonzero vector x that satisfies Ax = 0.</p> Signup and view all the answers

<p>Vectors are linearly independent if their linear combination has only the trivial solution.</p> Signup and view all the answers

<p>Vectors are linearly dependent if a linear combination has a nontrivial solution.</p> Signup and view all the answers

<p>A rule that assigns each vector x in R^n to a vector T(x) in R^m, satisfying specific properties.</p> Signup and view all the answers

<p>The range is the set of all vectors in R^m of the form T(u) where u is in R^n.</p> Signup and view all the answers

<p>The kernel is the set of all vectors u in R^n such that T(u) = 0.</p> Signup and view all the answers

<p>The column space is the set of all linear combinations of the columns of a matrix.</p> Signup and view all the answers

<p>The null space is the set of all solutions to the homogeneous equation Ax = 0.</p> Signup and view all the answers

<p>A unique matrix A exists such that T(x) = Ax for all x in R^n.</p> Signup and view all the answers

<p>T is one-to-one if each b in R^m is the image of at most one x in R^n.</p> Signup and view all the answers

<p>T is onto R^m if each b in R^m is the image of at least one x in R^n.</p> Signup and view all the answers

<p>The diagonal consists of all the entries in the aii positions of the matrix.</p> Signup and view all the answers

<p>A diagonal matrix is a square nXn matrix whose non-diagonal entries are zero.</p> Signup and view all the answers

<p>An mXn matrix whose entries are all zero is a zero matrix.</p> Signup and view all the answers

<p>An upper triangular matrix is formed when all entries below the diagonal are zero.</p> Signup and view all the answers

<p>A lower triangular matrix is formed when all entries above the diagonal are zero.</p> Signup and view all the answers

<p>The identity matrix has 1s on the main diagonal and zeros elsewhere, serving as the multiplicative identity.</p> Signup and view all the answers

<p>Associative, distributive, identity, and scalar multiplication properties govern the behavior of matrix multiplication.</p> Signup and view all the answers

<p>If A is an nXn matrix and k is a positive integer, then A^k denotes the product of k copies of A.</p> Signup and view all the answers

<p>The transpose of an mXn matrix A is the nXm matrix whose columns are formed from the corresponding rows of A.</p> Signup and view all the answers

<p>An nXn matrix A is invertible if there exists a matrix C such that CA = I and AC = I.</p> Signup and view all the answers

<p>A singular matrix is not invertible, typically characterized by a determinant of zero.</p> Signup and view all the answers

<p>A non-singular matrix is invertible, meaning its determinant is not zero.</p> Signup and view all the answers

<p>Multiply the inverse matrix on the left on both sides of the equation Ax = b.</p> Signup and view all the answers

<p>Augment the identity matrix with the original matrix to obtain the inverse.</p> Signup and view all the answers

<p>A vector space is a nonempty set of objects called vectors, defined by addition and scalar multiplication.</p> Signup and view all the answers

<p>A subspace is a subset of a vector space that includes the zero vector, closure under addition, and closure under scalar multiplication.</p> Signup and view all the answers

<p>A basis is a linearly independent set that spans the vector space.</p> Signup and view all the answers

<p>The standard basis is the set of columns of the nXn identity matrix.</p> Signup and view all the answers

<p>The dimension is the number of basis elements for the subspace.</p> Signup and view all the answers

<p>The rank of A is the dimension of the column space of A.</p> Signup and view all the answers

<p>The rank of a linear transformation L is the dimension of its image.</p> Signup and view all the answers

<p>The nullity is the dimension of the null space.</p> Signup and view all the answers

<p>The nullity is the dimension of the kernel where the transformation equals zero.</p> Signup and view all the answers

<p>The coordinates are the weights such that x = c1b1 + ... + cnbn.</p> Signup and view all the answers

<p>For n &gt;= 2, the determinant is the sum of n terms formed from the entries of the matrix with alternating signs.</p> Signup and view all the answers

<p>Cofactor expansion can be carried out across any row or down any column using the corresponding cofactors.</p> Signup and view all the answers

<p>Row operations affect the determinant in specific ways, like changing signs or scaling.</p> Signup and view all the answers

<p>The determinant of a diagonal matrix is the product of its diagonal elements.</p> Signup and view all the answers

<p>The determinant is the product of the diagonal elements.</p> Signup and view all the answers

<p>The determinant is found by similarly taking the product of the diagonal elements.</p> Signup and view all the answers

<p>Determinants have various properties, including associativity, sign changes with row swaps, and scaling.</p> Signup and view all the answers

<p>Cramer's rule provides a formula for unique solutions of Ax = b using determinants.</p> Signup and view all the answers

<p>Determinants can be used to calculate the area of geometric figures represented by vectors.</p> Signup and view all the answers