Linear Algebra with Applications 5th. Edition by Otto Bretscher | 9780321796974 | 321796977 | Holooly

Linear Algebra with Applications

179 SOLVED PROBLEMS

Question: 4.3.2

Use Theorem 4.3.2 to find the matrix B of the linear transformation T ( f ) = f + f from P2 to P2 with respect to the standard basis B = (1, x, x 2); see Example 1. ...

Verified Answer:

By Theorem 4.3.2, we have

B=\begin{bmatrix}...

Question: 6.2.6

Use Laplace expansion to compute det A for A=[1 0 1 2 9 1 3 0 9 2 2 0 5 0 0 3]. ...

Verified Answer:

Looking for rows or columns with as many zeros as ...

Question: 6.2.4

Find det[0 7 5 3 1 1 2 1 1 1 2 −1 1 1 1 2]. ...

Verified Answer:

We go through the elimination process, keeping a n...

Question: 6.2.2

Consider the matrices A[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9] and B=[6 7 8 9 8 1 2 3 4 5 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9]. Note that B is obtained from A by swapping the first two rows. Express det B in terms of det A. ...

Verified Answer:

For each pattern P in A, we can consider the corre...

Question: 6.2.1

Let A=[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9]. Express det(A^T ) in terms of det A. You need not compute det A. ...

Verified Answer:

For each pattern P in A, we can consider the corre...

Question: 6.1.11

Find det M for M=[a11 a12 b11 b12 a21 a22 b21 b22 0 0 c11 c12 0 0 c21 c22]. ...

Verified Answer:

It is natural to partition the 4 × 4 matrix M into...

Question: 6.1.9

Find det A for A=[6 0 1 0 0 9 3 2 3 7 8 0 3 2 9 0 0 4 0 0 5 0 5 0 1]. ...

Verified Answer:

Again, let’s look for patterns with a nonzero prod...

Question: 6.1.8

Find det A for A=[0 2 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0]. ...

Verified Answer:

Only one pattern P makes a nonzero contribution to...

Question: 7.4.5

Consider the linear transformation T f (x) = f (2x − 1) from P2 to P2. Is transformation T diagonalizable? If so, find an eigenbasis B and the B-matrix B of T . ...

Verified Answer:

Here it would be hard to find eigenvalues and eige...

Question: 7.6.3

Consider the dynamical system x(t + 1) =[3 −5 1 −1] x(t) with initial state x0 =[0 1]. Use Theorem 7.6.3 to find a closed formula for x(t), and sketch the trajectory. ...

Verified Answer:

In Example 8 of Section 7.5, we found the eigenval...

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