Linear Algebra Final Exam Flashcards

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Questions and Answers

What is a system of equations consistent?

It has only one solution.

When are the set of vectors S = {V1,V2,..,Vn} linearly dependent?

We can find a solution to X1V1 + X2V2 + .. + XnVn = 0 where not all coefficients equal 0.

When are the set of vectors S = {V1,V2,..,Vn} linearly independent?

Whenever X1V1 + X2V2 + ... + XnVn = 0 then X1 = X2 = ... = Xn = 0.

What is an eigenvector of an nxn matrix A?

A nonzero vector x satisfying A(x) = λx for some scalar λ. Signup and view all the answers

What does it mean for the set of vectors S = {V1,V2,..,Vn} to span a vector space V?

Every vector v ∈ V is a linear combination of V1,V2,..,Vn. Signup and view all the answers

What is a linear combination of the vectors V1,V2,..,Vn?

There exist scalars x1,x2,..xn satisfying V = x1V1 + x2V2 + ... + xnvn. Signup and view all the answers

What characterizes a linear transformation T: R → R?

It satisfies a.T(x+y) = T(x) + T(y) and b.T(cx) = cT(x) for all vectors x,y ∈ R and scalars c ∈ R. Signup and view all the answers

What are the three elementary row operations?

a.Replace one row with the sum of itself and a multiple of another row, b.Interchange 2 rows, c.Multiply all entries in the row by a nonzero constant. Signup and view all the answers

What are the properties of a rectangular matrix in reduced echelon form?

a.All nonzero rows above all-zero rows, b.Leading entry of each row right of the above row's leading entry, c.Leading entry in each nonzero row is 1, d.Leading 1 is the only nonzero entry in its column. Signup and view all the answers

What defines a scalar λ as an eigenvalue of an nxn matrix A?

A(x) = λx for some nonzero vector x. Signup and view all the answers

What is the null space of a linear transformation T: R → R?

Nul T = {x ∈ R: T(x) = 0}. Signup and view all the answers

When is a function f: x → y considered onto?

For each y ∈ Y, there is at least one x ∈ X with f(x) = y. Signup and view all the answers

When is a function f: x → y considered 1-1?

Whenever f(r) = f(s), then r = s. Signup and view all the answers

What is the characteristic polynomial p(λ) for an nxn matrix A?

p(λ) = det(A - λI). Signup and view all the answers

How is the norm of a vector v = [x1,x2,..,vn] defined?

||v|| = √(x1² + x2² + .. + xn²). Signup and view all the answers

What is the eigenspace of A corresponding to eigenvalue λ?

The set of all solutions to det(A - λI) = 0. Signup and view all the answers

When are two nxn matrices A and B considered similar?

There is an invertible nxn matrix P such that A = PBP⁻¹. Signup and view all the answers

What does it mean for a nxn matrix to be diagonalizable?

A = PDP⁻¹ where D is a diagonal matrix and P is an invertible nxn matrix. Signup and view all the answers

When are two vectors u and v considered perpendicular (or orthogonal)?

Their dot product is zero. Signup and view all the answers

When is a set of vectors B = {V1,V2,..Vn} a basis for a vector space V?

B is a linearly independent set of vectors and B spans V. Signup and view all the answers

Study Notes

Systems of Equations

A system is consistent if it has only one solution.

Linear Dependence and Independence

Vectors are linearly dependent if a linear combination equals zero with at least one non-zero coefficient.
Vectors are linearly independent if the only way a linear combination equals zero is when all coefficients are zero.

Eigenvectors and Eigenvalues

An eigenvector x of an nxn matrix A satisfies A(x) = λx for some scalar λ.
A scalar λ is an eigenvalue of A if there exists a nonzero vector x such that A(x) = λx.

Vector Spaces and Spanning

A set of vectors spans a vector space V if every vector in V can be expressed as a linear combination of these vectors.

Linear Combinations

A vector V is a linear combination of vectors V1, V2, ..., Vn if there are scalars x1, x2, ..., xn such that V = x1V1 + x2V2 + ... + xnvn.

Linear Transformations

A function T: R → R is a linear transformation if it satisfies:
- T(x+y) = T(x) + T(y) for all x, y ∈ R.
- T(cx) = cT(x) for all x ∈ R and scalars c ∈ R.

Elementary Row Operations

The three elementary row operations on matrices are:
- Replacing one row with the sum of itself and a multiple of another row.
- Interchanging two rows.
- Multiplying all entries in a row by a nonzero constant.

Reduced Echelon Form

A matrix is in reduced echelon form if:
- All nonzero rows precede any row of zeros.
- Leading entries appear in a column to the right of leading entries in rows above.
- Each leading entry is 1.
- Each leading 1 is the only nonzero entry in its column.

Null Space

The null space Nul T of a linear transformation T: R → R consists of vectors x ∈ R such that T(x) = 0.

Onto and One-to-One Functions

A function f: X → Y is onto if for each y ∈ Y, there exists at least one x ∈ X with f(x) = y.
A function f: X → Y is one-to-one (1-1) if f(r) = f(s) implies r = s for any r, s ∈ X.

Characteristic Polynomial

The characteristic polynomial p(λ) of an nxn matrix A is defined as p(λ) = det(A - λI).

Vector Norm

The norm of a vector v = [x1, x2, ..., xn] is defined by ||v|| = √(x1² + x2² + ... + xn²).

Eigenspace

The eigenspace corresponding to an eigenvalue λ is the set of all solutions to the equation det(A - λI) = 0.

Similar Matrices

Two nxn matrices A and B are similar if there exists an invertible nxn matrix P such that A = PBP⁻¹.

Diagonalizable Matrices

An nxn matrix is diagonalizable if it can be expressed as A = PDP⁻¹, where D is a diagonal matrix and P is an invertible matrix.

Orthogonal Vectors

Two vectors u and v are orthogonal if their dot product equals zero.

Basis for a Vector Space

A set of vectors B = {V1, V2, ..., Vn} is a basis for a vector space V if B is linearly independent and spans V.

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Description

Test your knowledge on key concepts for your Linear Algebra final exam. This quiz features essential definitions such as consistent systems of equations and linear dependence and independence of vectors. Review and prepare effectively for your upcoming assessment!