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nehap.2491
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suppose that vectors in R3 are denoted by 1*3 matrices, and define T:R4 to R3 by T9x,y,z,t)=(x-y+z+t,2x-2y+3z+4t,3x-3y+4z+5t).Find basis of kernel and range.
Thank you!micromass said:Ow, for the image you won't need to do all that stuff, I'm sorry.
You'll first have to find a basis of R4, call this {e1,e2,e3,e4}. Then {T(e1),T(e2),T(e3),T(e4)} is a set which spans the image. If this set is linear independant, then it's a basis. If not, then remove some vectors until it is linear independant...
Thank you!Outlined said:If you are open to it, the equation below (in case f : V -> W) might even help:
dim(ker(f)) + dim(Im(f)) = dim(V)
A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the linear structure of the original space.
To find the basis of a kernel, you can set up a system of equations using the matrix representation of the linear transformation and solve for the variables. The basis of the kernel will be the set of vectors that satisfy the system of equations when plugged into the transformation.
The range of a linear transformation is the set of all possible output vectors that can be obtained by applying the transformation to all possible input vectors. In other words, it is the span of the columns of the matrix representation of the transformation.
A linear transformation is one-to-one if and only if its kernel contains only the zero vector. This means that no two distinct input vectors can produce the same output vector.
Yes, a linear transformation can have a non-trivial kernel and range. This means that there are input vectors that produce a zero output vector and there are output vectors that cannot be obtained by applying the transformation to any input vector, respectively.