# Change of Basis- Please help

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• May 17th 2009, 04:35 AM
SydneyBristow
Change of Basis- Please help
(Headbang)

Please help me, i just dont get change of basis. The book omits the steps and i dont understand the notation to understand what exactly i am supposed to do.

This is the example in the book, i dont understand how they arrive at the solution because i dont understand what the notation means and it is driving me insane already because i am sure that the actual calculations are easy.

x1+x2+x3=0 is a plane, consider the bases U and B in this plane.

U= (a1,a2)= (0,1,-1), (1,0,-1)
B= (b1,b2) = ( 1,2,-3),(4,-1,-3)

Find the change of basis matrix.

The book then does

[b1]subscript U= [2,1]

Also if I have diagnolized a matrix how do i find the change of basis matrix to go from the original to the diagnol?
• May 18th 2009, 12:46 AM
scorpion007
Quote:

Originally Posted by SydneyBristow
U= (a1,a2)= (0,1,-1), (1,0,-1)
B= (b1,b2) = ( 1,2,-3),(4,-1,-3)

Find the change of basis matrix.

The book then does

[b1]subscript U= [2,1]

What they mean by \$\displaystyle {b_1}_U\$ is likely \$\displaystyle b_1\$ relative to the \$\displaystyle U\$ basis.

I.e. \$\displaystyle b_1 = 2a_1 + 1a_2 = (2,1)\$ relative to \$\displaystyle U\$.

Remember, \$\displaystyle U\$ and \$\displaystyle B\$ are both bases for the same subspace of \$\displaystyle \mathbb{R}^3\$, so any vector in \$\displaystyle B\$ can be expressed as a linear combination of the vectors in \$\displaystyle U\$, and vice versa. In doing so, you get its co-ordinates relative to that basis.