Kernel & Image of Linear Transformation Homework

In summary, the conversation discusses determining whether or not two given vectors, v1 and v2, are in the kernel of a linear transformation T:R^4 > R^3 represented by a given matrix A. The solution involves row-reducing A and solving for x1, x2, x3, and x4, and concluding that v1 and v2 are not in the kernel. The question also asks to determine whether or not two other vectors, w1 and w2, are in the image of the same linear transformation, but the solution is not provided. To determine if a vector is in the kernel, take the product of the matrix and the vector, which should result in a zero vector if the vector is
  • #1
jeffreylze
44
0

Homework Statement



38) Determine whether or not v1 = (-2,0,0,2) and v2 = (-2,2,2,0) are in the kernel of the linear transformation T:R^4 > R^3 given by T(x) = Ax where

A = [1 2 -1 1;
1 0 1 1;
2 -4 6 2]

39) Determine whether or not w1 = (1,3,1) or w2 = (-1,-1,-2) is in the image of the linear transformation given in question 38?

Homework Equations


The Attempt at a Solution



I row-reduced it to rref then i let matrix = 0 and then solve for x1, x2, x3 and x4 . Which gave me x3 (-1,1,1,0) + x4 (-1,0,0,1) . v1 and v2 are not in that kernel i found but the answer states otherwise. is it because v1 and v2 are just scalar multiples of x3 and x4 ?For question 39 , i am stuck too. Please help.
 
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  • #2
To determine whether or not given vectors are in the kernel of a linear transformation, simply take the product of the matrix which represents the linear transformation and the vector in question.

What should this product be if the vector is in the kernel?
 
  • #3
The product should give me a zero vector. What about for the image, question 39 ?
 

Related to Kernel & Image of Linear Transformation Homework

1. What is a kernel in linear transformation?

The kernel of a linear transformation refers to the set of all vectors in the domain that are mapped to the zero vector in the codomain. In other words, it is the set of all inputs that result in an output of zero. The kernel is a subspace of the domain and is also known as the null space.

2. How is the kernel related to the image of a linear transformation?

The kernel and the image of a linear transformation are complementary subspaces. The image is the set of all outputs that are actually reached by the transformation, while the kernel is the set of all inputs that are not reached. In other words, the kernel and image together make up the entire domain.

3. Why is the kernel important in linear transformation?

The kernel is important because it allows us to understand the behavior of a linear transformation. It gives insight into which inputs will result in a zero output and which inputs will actually be transformed. The dimension of the kernel also tells us about the invertibility of a linear transformation.

4. How do you find the kernel of a linear transformation?

To find the kernel of a linear transformation, we set up a system of equations using the standard matrix representation of the transformation. We then use row reduction techniques to solve for the free variables, which will give us the basis for the kernel. Alternatively, we can also use the null space function in a computer algebra system to find the kernel.

5. Can a linear transformation have a non-trivial kernel?

Yes, a linear transformation can have a non-trivial kernel. In fact, most linear transformations do have a non-trivial kernel. This means that there are inputs that will result in a zero output. Only the zero transformation, which maps all inputs to the zero vector, has a trivial kernel.

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