Exam questions

Question 1

matrix with some unknown values
find values of k for which Ax=b has
1- no solution
2- a unique solution
3-infinitely many solutions

Answer 1

row reduce [A b] to reduced echelon form
1- find k such that one row is [0 0 0 c] (inconsistent)
2- find k in expression a such that a /= 0 [0 0 a c]
3- find k for a = 0 and c = 0

Question 2

Are the columns of B linearly dependent?

Answer 2

row reduce B to reduced echelon form

• pivot in every column : linearly independent
• no pivot in every column : linearly dependent

Question 3

Find basis for
1- col A
2- Row A
3- Nul A

Answer 3

row reduce to reduced echelon form
1- basis = colums of original matrix where there is pivot positions
2- basis = rows of original matrix where there is pivot positions
3- solve Bx=0 x=x3(a,b,c) then basis is the vector abc

Question 4

Given a system, use Cramer’s rule to find the value of one of the variables (z)

Answer 4

take A as the matrix with the left hand side of the equations in the system
take b the vector with the right hand side of the equations
compute det A
compute det Ai(b) such that the value in i is the variable
use Cramer’s rule : z=(det Ai(b))/det A

Question 5

Given two matrices A and C

Find the matrix B that solves AB= C

Answer 5

AB=C <=> B=A^-1 C
compute A^-1 with A^-1 = (1/det A) * adj A
compute B = A^-1 C
OR
row reduce [A C]

Question 6

Given two bases A and B

Find the chnage-of-coordinates matrix from A to B

Answer 6

row reduce to reduced echelon form [B /A]
obtain [I/C]
C is P(C

Question 7

Given a set H

Prove that H is a subspace of Rn

Answer 7

show that 
-zero vector is in H
-u+v is in H
-cu is in H
OR
show H is a span of vectors