Question

linear algebra help

Answer 1

@amistre64

Answer 2

to show 1) A+B=B+A
let A be (a11 a12 ; a21 a22) (that is supposed to be a 2 x 2 matrix)
and B be (b11 b12; b21 b22)
now add the two matrices (you add corresponding elements)
you get
A+B= (a11 + b11 a12+b12; a21+b21 a22+b22)

each entry in the the matrix can be "swapped" because addition is commutative
so we can say
= (b11+a11 b12+a12; b21+a21 b22+a22)
and that is the same as B+A

Answer 3

ok so wat about the 2nd property how would i do that?

Answer 4

let A be (a11 a12 ; a21 a22)
what would you add to A to get 0's in every location?

Answer 5

(-a11 -a12 -a21 -a22)?

Answer 6

yes, and that works for all possible a's
so A^(-1) in this problem would be -A or A with each entry negated

Answer 7

ok and tthe third property?

Answer 8

if you multiply a matrix by a scalar , you multiply each entry in A by that scalar
in other words
b A = [ b*a11 b*a12; b*a21 b*a22]
and of course, vice versa:
[ b*a11 b*a12; b*a21 b*a22] = b*A

Answer 9

so show (ab) which is a scalar, times A gives
a*b*a11 a*b*a12; a*b*a21 a*b*a22

now do
b*A
and then multiply that result by a
you should get the same as the line up above, showing
(ab)*A = a*(b*A)

(btw the scalar a is not the same as the entries a11, etc in the matrix)

Answer 10

I am naming the entries in matrix A \(a_{11}\) etc
(I have to call them something)
but your problem says show
(ab)A
and their a is just some number.
if you like, change the name of the numbers in A to \(c_{11}, c_{12}\) , etc

Answer 11

the idea is
(ab) means multiply a*b to get a number
and then multiply each entry in matrix A by that number
in other words we would say (ab) is (a*b)
and multiply each entry by (a*b)
then drop the parens, so each entry would be \( a\cdot b \cdot c_{11} \) etc

Answer 12

ok i understand and when u do it for b*a u shd get the same right ?

Answer 13

on the other hand
(bA)
means first multiply b times each entry in A. You should get \(b\cdot c_{11}\) etc
then multiply that new matrix by a

Answer 14

ok i have 1 more question

Answer 15

They are asking if you can add any two members of S_2, and the result is also in S_2
or if we multiply by a scalar, the result is in S_2
when they say y=-x are positive, I think that means x is negative
and the members in S_2 are (-1,1), (-2,2) , (-3,3), (-1.111, 1.111), etc

Answer 16

I can see a problem if we multiply by the scalar -1
for example
-1 * (-1,1) becomes (1,-1)
and y is -1 and not a positive number. In other words when we multiply by -1, we get a result that does not meet the definition of S_2. That means we do not have a subspace

Answer 17

o ok so its not a subspace of V

Answer 18

yes, it is not a subspace of V