###### 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 ]) ...

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 ...

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. ...

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, ...

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. ...

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] ...

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 ] ...

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 ...

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 ...

Since T (0) = T\left(\begin{bmatrix} 0 \\ 0...
First observe that if p(x) is in  P_{3}[/la...