Linear Algebra Answers

Use the properties of determinants to evaluate the following determinant:

(b+c)2 a2 a2
b2 (c+a)2 b2
c2 c2 (a+b)2

Let B1={[1 1], [1 0], [1 0] and B2={[1 0], [0 1], [0 1]
[0 1] [0 1] [0 -1] } [0 0] [0 1] [0 -1] }
be 2 bases for span(B1) in M22, with the usual left to right ordering.
Let B3 be a basis for P1 and be the transition matrix from B2 to B3 giiven by [1 1 1]
[0 1 1] = PB2→B3
[0 0 1]
a) Find transition matrix PB1→B3
b) Use PB2→B3 to find B3

Standard basis vectors for R^3 are (1,0,0),(0,1,0) and (0,0,1). If we want to insert u → into this basic, then which vector from standard basis can be removed while still maintaining the basis of R^3.
Discuss the case when:
u → =(4,3,6)
u → =(4,0,6)
Interpret the result geometrically in both cases.

which sets are a basis for the following vector subspace of P2 :
X={A e M22 : A [1] = [0] }
[2] [0]

A {[2 0] , [0 -1] , [0 0] , [0 0] } C {[2 -1] , [0 0] }
[0 0] [0 0] [2 0] [0 -1] [0 0] [2 -1]

B{[2 -1] } D { [2 -1] , [2 -1] }
[2 -1] [2 -1] [-2 1]

determine if the set is a subspace of r2 or not

{ x: (x1,x2) such that x1= -x2 }

is this a subspace ?
.

Solve the following matrix by Gaussian Model

1.) -4x + y = 3
3x - 5y = 0


2.) 3x + 4y + 5z = -9
2x - 3y + 3z = -1
x + 2y - 4z = -15

Solve the matrix of the following using the Gaussian Method.

1. 3x-5y= -2
2x+4y= 7

Show that the vectors (1-i,i) and (2,-I+i) in R² are Linearly Dependent over Field R but Linearly Independent over R , where i = √-1

Show that the vectors (1-I,i) and (2,-I+i) in R² are Linearly Dependent over Field but Linearly Independent over R , where .i = √-1

Let A be an n×n matrix. Then Show that the set, U={uϵRn : Au = - 3un} is a Subspace ofRn