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?

<p>$A$ is a singular matrix (A)</p> Signup and view all the answers

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

<p>$C1u + C2v = 0$ with $C1 \neq 0$ and $C2 \neq 0$ (C)</p> Signup and view all the answers

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

<p>They must include the zero vector (A)</p> Signup and view all the answers

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

<p>$\begin{bmatrix} 1 &amp; 2 &amp; -1 \ 0 &amp; 0 &amp; 0 \ 0 &amp; 0 &amp; -4 \end{bmatrix}$ (C)</p> 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?

<p>h-3, -5, 3i (A)</p> 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?

<p>h11, 8i (C)</p> Signup and view all the answers

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

<p>h3, -2, 4i (A)</p> 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?

<p>They are inverse matrices. (C)</p> 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?

<p>$n × n$ (D)</p> 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 $0 \leq \alpha \leq \frac{\pi}{2}$?

<p>$\begin{bmatrix} \cos(\alpha) &amp; -\sin(\alpha) \ \sin(\alpha) &amp; \cos(\alpha) \end{bmatrix}$ (D)</p> 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?

<p>$2i + j + 4k$ (C)</p> 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?

<p>$\begin{bmatrix} 3 &amp; 0 &amp; 0 \ 0 &amp; 3 &amp; 0 \ 0 &amp; 0 &amp; 3 \end{bmatrix}$ (B)</p> Signup and view all the answers

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

<p>No, because a projection onto a line may not be invertible. (D)</p> 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?

<p>Remains unchanged (C)</p> 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?

<p>$\begin{bmatrix} -1 &amp; 0 &amp; 0 \ 0 &amp; -1 &amp; 0 \ 0 &amp; 0 &amp; -1 \end{bmatrix}$ (B)</p> 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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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.

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