Linear Transformation and Eigenvalues Quiz

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Questions and Answers

Which of the following best defines a linearly dependent set of vectors in R3?

They are orthogonal to each other

They lie in the same plane (correct)

Their tails are in the same direction

They form the sides of a triangle

What is a square matrix called if it has an inverse?

Non-invertible

Singular

Indeterminate

Non-singular (correct)

In a linearly dependent set, what condition must be satisfied for two vectors u and v?

$C1u + C2v = 0$ with $C1 \neq 0$ and $C2 \neq 0$

$u = -v$

$u = v$

$u = \lambda v$ (correct)

What property characterizes a square matrix A that is not invertible?

$A$ is a singular matrix Signup and view all the answers

What is the condition for vectors u and v to be linearly dependent according to the text?

$C1u + C2v = 0$ with $C1 \neq 0$ and $C2 \neq 0$ Signup and view all the answers

What is required for a set of vectors to be considered linearly dependent according to the passage?

They must include the zero vector Signup and view all the answers

What is the matrix associated with the linear transformation T given in Example L29-5?

$\begin{bmatrix} 1 & 2 & -1 \ 0 & 0 & 0 \ 0 & 0 & -4 \end{bmatrix}$ Signup and view all the answers

What is the image of the vector h1, 2, -1i under the linear transformation T described in Example L29-5?

h-3, -5, 3i Signup and view all the answers

Given T(u) = Au and A = $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, what is the linear transformation T's action on the vector h1, 0i?

h11, 8i Signup and view all the answers

If T(u) = h5, -3, 12i, what is the vector u such that T(u) = h5, -3, 12i?

h3, -2, 4i Signup and view all the answers

In inverse transformations, if S is the inverse transformation of T, what is the relationship between T's matrix A and S's matrix B?

They are inverse matrices. Signup and view all the answers

If A and B are matrices associated with T and T^(-1) respectively, what size are these matrices when T is a transformation from R^n to R^n?

$n × n$ Signup and view all the answers

What is the matrix associated with the linear transformation that rotates every vector counterclockwise about the origin by an angle of $\alpha$, where $0 \leq \alpha \leq \frac{\pi}{2}$?

$\begin{bmatrix} \cos(\alpha) & -\sin(\alpha) \ \sin(\alpha) & \cos(\alpha) \end{bmatrix}$ Signup and view all the answers

For a linear transformation T that reflects vectors in the yz-plane and then doubles their length, what is the image of the vector $v = i - j + 2k$ under T?

$2i + j + 4k$ Signup and view all the answers

Which of the following matrices represents the linear transformation that projects every vector onto the xy-plane and then triples its length in R3 space?

$\begin{bmatrix} 3 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 3 \end{bmatrix}$ Signup and view all the answers

Does the linear transformation T that projects every vector onto the line $y = x$ have an inverse?

No, because a projection onto a line may not be invertible. Signup and view all the answers

In finding the matrix of a linear transformation T: R2 → R2 that rotates vectors clockwise about the origin by an angle of $\frac{\pi}{3}$, what happens to the determinant of A − λI if you replace the first row with the sum of all rows?

Remains unchanged Signup and view all the answers

For a linear transformation T that reflects vectors in the yz-plane and then doubles their length, what is the matrix associated with its inverse transformation?

$\begin{bmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{bmatrix}$ Signup and view all the answers

Study Notes

Linear Transformations and Matrices

A linear transformation T : R2 → R2 that projects every vector onto the line y = x has a specific matrix.
A linear transformation T : R3 → R3 that reflects every vector in the yz-plane and then doubles its length has a specific matrix.

Eigenvalues and Eigenvectors

Let T : Rn → Rn be a linear transformation.

Geometry behind Linear Dependence of Vectors

Two non-zero vectors u and v are linearly dependent if and only if they are parallel, i.e., u = λv, for some λ ∈ R.
Three non-zero vectors in R3 are linearly dependent if and only if they lie in the same plane when their tails are put together.

Inverse Matrices

A square matrix A is called invertible (or non-singular) if there exists a matrix denoted by A−1 such that A−1 A = AA−1 = I.
Matrix A−1 is called the inverse of A.
If A does not have an inverse, it is called singular.

Matrices for Linear Transformations

A linear transformation T has a matrix associated with it.
The matrix associated with T can be found using different approaches.

Inverse Transformations

If a linear transformation T : Rn → Rn has an inverse T −1 , then there exists another linear transformation S : Rn → Rn that reverses the action of T , i.e., if T (u) = v, then S(v) = u, for any u ∈ Rn.
Suppose that A and B are matrices associated with T and T −1 , respectively.

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Description

Test your understanding of finding the matrix of a linear transformation and determining eigenvalues and eigenvectors. Includes examples of projecting vectors onto specific lines and reflecting vectors in certain planes.