Introduction to Linear Algebra with Applications 1st. Edition by James DeFranza, Daniel Gagliardi | 9780073532356 | 0073532355

Linear Algebra

Introduction to Linear Algebra with Applications

183 SOLVED PROBLEMS

Question: 4.1.9

Let T: R³→ R³ be a linear operator and B a basis for R³ given by B = { [ 1 1 1 ], [ 1 2 3 ], [ 1 1 2 ] } If T([ 1 1 1 ])= [ 1 1 1 ] T([ 1 2 3 ])= [- 1 -2 -3 ] T([ 1 1 2 ])= [2 2 4 ] find T([ 2 3 6 ]) ...

Verified Answer:

Since B is a basis for R³, there are (unique) scal...

Question: 5.1.3

Find the eigenvalues of A= [ 1 0 0 0 0 1 5 -10 1 0 2 0 1 0 0 3] and find a basis for each of the corresponding eigenspaces ...

Verified Answer:

The characteristic equation of A is

det(A −...

Question: 4.3.4

Find an explicit isomorphism from P2 onto the vector space of 2 × 2 symmetric matrices S2×2. ...

Verified Answer:

To use the method given in the proof of Theorem 11...

Question: 4.1.7

Coordinates Let V be a vector space with dim(V ) = n, and B = {v1, v2, . . . , vn} an ordered basis for V. Let T: V → R^n be the map that sends a vector v in V to its coordinate vector in R^n relative to B. That is, T (v) = [v]B It was shown in Sec. 3.4 that this map is well defined, that is, ...

Verified Answer:

Let u and v be vectors in V and let k be a scalar....

Question: 5.3.1

Find the general solution to the system of differential equations{ y′1= −y1 y′2= 3y1 + 2y2 Sketch several trajectories in the phase plane. ...

Verified Answer:

The differential equation is given in matrix form ...

Question: 4.3.3

Let T: R²→R² be the mapping of Example 1 with T (v) = Av, where A= [ 1 1 -1 0] Verify that the inverse map T−1:R² → R² is given by T ^−1(w) = A^−1w, where A^-1= [0 -1 1 1] ...

Verified Answer:

Let v =

\begin{bmatrix} v_{1}\\ v_{2} \end...

Question: 4.1.8

Let T: R³ → R² be a linear transformation, and let B be the standard basis for R³. If T (e1) [ 1 1 ] T (e2) = [ -1 2 ] and T (e3) = [ 0 1 ] find T (ν), where ν = [ 1 3 2 ] ...

Verified Answer:

To find the image of the vector

ν

, ...

Question: 4.1.6

Define a mapping T: Mm×n → Mn×m by T (A) = A^t Show that the mapping is a linear transformation ...

Verified Answer:

By Theorem 6 of Sec. 1.3, we have

T (A+ B) ...

Question: 4.1.5

Define a mapping T: R²→ R² by T([ x y ])= [e^x e^y ] Determine whether T is a linear transformation ...

Verified Answer:

Since

T (0) = T\left(\begin{bmatrix} 0 \\ 0...

Question: 4.1.4

Define a mapping T: P3 → P2 by T (p(x)) = p′(x)where p′(x) is the derivative of p(x).a. Show that T is a linear transformation. b. Find the image of the polynomial p(x) = 3x^3 + 2x^2 − x + 2. c. Describe the polynomials in P3 that are mapped to the zero vector of P2. ...

Verified Answer:

First observe that if p(x) is in

P_{3}[/la...