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Algebra
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10 Nov 2019

1. Consider the matrix A=[ 1 1 1

1 1 1

1 1 1]

a) Find the eigenvalues of A.

b) Determine an invertible matrix P and adiagonal matrix D such that P-1AP=D.

c) Compute A^100.

(may include c^100 for scalar c)(need toexplain.)


2. Which is a subspace? (Don't needexplain.)

A. {[x

y

z]:x-y+z=1} is a subspace of R^3.

B. {[-s+3t

s-2t

5s-t]: s,tbelong to R} is a subspace of R^3

C. {[s+1

-t

s+t]: s,t belong to R}is a subspace of R^3

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Irving Heathcote
Irving HeathcoteLv2
28 Jan 2019

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2.15

1) solving the following system of linear equations:

x1+ 2x2 + 3x3 + 4x4 + 5x5= 6

x1+ 2x2 + 4x3 + 3x4 + 7x5= 5

x1+ 2x2 + 2x3 + 5x4 + 4x5= 9

Write the solution in parametric form.


2) Define a function T:R^(3) to R^(2) by T(x,y,z) = (x + y + z,x +2y - 3z)

a) Show that T is a linear transofrmation

b) Find all the vectors in the kernel of T

c) Show that T is onto

d) Find the matrix representation of T relative to the standardbasis of R^(3) and R^(2)


3) Show that B= {(1,1,1), (1,1,0), (0,1,1)} is a basis forR^(3). Find the coordinate vector of (1,2,3) relative to the basisof B.


4) Use the Gram-Schmidt process to find an orthonormal basis forthe subspace of R^(4) spanned by the vectors(1,0,1,1), (1,0,1,0), (0,0,1,1).


5) Consider the Matrix B = {(3,2,-3), (-3,-4,9), (-1,-2,5)}.Find an invertible matrix P and a diagonal matrix D such thatB=PDP^(-1).


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