Finding Basis for R2 Using Linear Transformation with Transition Matrix P s←t

[jlucas134]

Need some help getting started...

Let T ={ [1, 0], [1, 1] }be a basis for R2 .
Given that Transition matrix P s←t
[ 2, 3 ; -1, 2],

find the basis S for R2.

Here is what I think...I started by letting v being any vector...

[1,0] and [0,1] and applied them to the transition matrix by multiplying the transition matrix to each individual set list at the beginning of this sentence to find the v value... to get [2,-1] and [3, 2]

Then taking these values to the vectors of T

I get 2*[1,0]-1*[1,1] and
3*[1,0]+2*[1,1]

My final answer came up with [1, -1] and [5, 2], so is this the basis of S?


2)
Let S ={ [1, -1], [1, 1] } be a basis for R2 .
Given that Transition matrix P s←t

[ 1, 2; 2, 3]

find the basis T for R2.

I think you have to inverse the trans matrix then do the steps from problem one...I get T={[-1,5], [1,-3]} for basis.

 

[mighty2000]

Hello there!

First of all, great job on starting to think about the problem and coming up with a solution. Let's take a closer look at your approach and see if we can refine it a bit.

In the first problem, you correctly started by letting v be any vector and applying the transition matrix to find the v value. However, instead of multiplying the transition matrix to each individual set, you should multiply it to each individual vector in the basis T. So, for [1,0], you would get [2, -1] and for [1,1], you would get [3,2].

Next, you took these values and applied them to the vectors in T. This is not the correct approach. Remember, a basis for a vector space is a set of linearly independent vectors that span the space. So, to find the basis S for R2, we need to find two linearly independent vectors that can be written as a linear combination of [2,-1] and [3,2].

Using the values we got earlier, we can write [1,-1] as 1*[2,-1] + (-1)*[3,2] and [5,2] as 2*[2,-1] + 1*[3,2]. So, the basis S for R2 is {[2,-1], [3,2]}.

For the second problem, you are correct in thinking that you need to find the inverse of the transition matrix. However, instead of applying it to the vectors in S, you should apply it to the vectors in the standard basis for R2, which is {[1,0], [0,1]}. This will give you the vectors in the basis T for R2 as {[1,2], [2,3]}.

I hope this helps and good luck with your further explorations in linear algebra!