The matrix A = < b m a t r i x > has characteristic...

The matrix $A = \begin{bmatrix} 2 & -1 \\ 1 & 4 \end{bmatrix}$ has characteristic polynomial $p(\lambda) = (\lambda - 3)^2$. 9.1 (5 pts) Find a basis for $E_{\lambda = 3}$. Show all your work and write the basis in the space provided. basis for $E_{\lambda = 3}$: $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$

Transcript

00:01 In this problem we are given the standard basis of the space p subter of polynomials as follows.

00:09 B equal to the set of 1 t, t squared and t cube.

00:16 So in the coordinate vector notation, we can write this as 1 .00, 0100 ,000, 0 ,000, 0 ,000, and 000, 00 ,000, and 0.

00:30 Zero zero one now using these coordinate vectors we are going to test the linear independence of the set of the polynomials given by three minus t cube minus two minus t squared and minus 23 plus 31 t minus 8 t squared plus t cube and then we are also going to write the coordinate vector for the polynomial 3 minus t cube namely this first term so we are going to write down the coordinate vectors for these polynomials anyway so let's get started with the second part actually okay let's call this set b2 for the sake of completeness.

01:35 Okay, let us expand these powers.

01:37 We have 27 minus 27 t plus 9 t squared minus t cube.

01:47 And 4 plus 4t plus t squared and also this last guy.

02:00 Okay, using this as standard basis vectors, we can re -express or equivalently expressed this b2 basis is 27 minus 27 9 minus 1 4 1 0 0 0 minus 23 1 minus 8 1 so i i have essentially read of the coefficients of named the constant term here the linear turn the quadratic turn and the cubic turn okay so let me just say that this is the coordinate vector for this polynomial 3 minus t cube okay with that out of the way let us show that these vectors are linearly independent i'm going to call them v1 v2 and v3 so how do you show that any set of vectors are linearly independent.

03:12 We form the following summation, some real number times v1, plus some other real number times v2, plus some other real number times v3.

03:25 This is set equal to 0, and we say that these vectors are linearly independent only if a equal to b equal to c equal to 0.

03:40 So we are going to write down the linear system of equations for this system and we will reduce the augmented matrix until we find the solutions a, b, and c.

03:53 If they are all equal to zero, we will conclude that these vectors are or these polynomials are linearly independent.

04:01 So let's do that.

04:05 Coefficient matrix is formed by writing these vectors side by side in a column manner.

04:12 So minus 27, 9 ,1, 4, 410, minus 23, 31, minus 8, 1.

04:25 And we have this vertical line and the right and side is 0.

04:29 Okay, this is the augmented matrix...

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