Question

Math

Posted 5 months ago

`Let V be the real vector space of all real 2 x 3 matrices, and let W be the real vector space of all real 4 x 1 column vectors. If T is a linear transformation from V onto W, what is the dimension of the subspace kernel of T?`

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

Determine the dimensions of the domain $V$ and codomain $W$

step 2

The dimension of $V$, the space of all real $2 \times 3$ matrices, is $2 \times 3 = 6$

step 3

The dimension of $W$, the space of all real $4 \times 1$ column vectors, is $4$

step 4

Apply the Rank-Nullity Theorem: $\text{dim}(V) = \text{dim}(\text{Kernel}(T)) + \text{dim}(\text{Image}(T))$

step 5

Since $T$ is onto $W$, the dimension of the image of $T$ is equal to the dimension of $W$, which is $4$

step 6

Substitute the known values into the Rank-Nullity Theorem: $6 = \text{dim}(\text{Kernel}(T)) + 4$

step 7

Solve for the dimension of the kernel of $T$: $\text{dim}(\text{Kernel}(T)) = 6 - 4$

step 8

Conclude that the dimension of the kernel of $T$ is $2$

Answer

The dimension of the subspace kernel of $T$ is $2$.

Key Concept

Rank-Nullity Theorem

Explanation

The Rank-Nullity Theorem relates the dimensions of the domain, kernel, and image of a linear transformation, which allows us to find the dimension of the kernel when the transformation is onto.

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