- #1
Tsunami317
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Homework Statement
Determine the standard matrix for the linear operator defined by the formula below:
T(x, y, z) = (x-y, y+2z, 2x+y+z)
Homework Equations
The Attempt at a Solution
No idea
HallsofIvy said:Because they say the "standard" matrix, they want you to use the "standard" basis, <1, 0, 0>, <0, 1, 0>, and <0, 0, 1>. What do you get when you multiply the matrix above by each of those?
You are told that T(x, y, z) = (x-y, y+2z, 2x+y+z). What is T(1, 0, 0)? What is T(0, 1, 0)? What is T(0, 0, 1)?
Compare those with the result of multiplying the matrix.
Tsunami317 said:So the answer would be
[x-y 0 0
0 y+2z 0
0 0 2x+y+z]
?
or is the answer
[x-y
y+2x
2x+y+z]
Tsunami317 said:I tried to multiply the matrix
1 0 0
0 1 0
0 0 1
by the vectors given but I got this
[x-y
y+2x
2x+y+z]
Doesn't make sense
A linear operator is a mathematical function that maps one vector space to another while preserving the linear structure and operations of the vector space.
A standard matrix for a linear operator is a matrix representation of a linear operator, where the columns of the matrix represent the images of the standard basis vectors for the input vector space.
To find the standard matrix for a linear operator, you can apply the linear operator to each of the standard basis vectors for the input vector space and then arrange the resulting vectors as columns in a matrix.
The standard matrix for a linear operator is useful because it allows us to easily perform calculations involving the linear operator, such as finding its inverse or determining its eigenvalues and eigenvectors.
Yes, the standard matrix for a linear operator can change depending on the basis chosen for the input and output vector spaces. However, the linear operator itself remains the same regardless of the chosen basis.