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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∻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.⚹◊What is the rank-nullity theorem and how does it apply to determining the dimension of the subspace kernel of a linear transformation $T$ from V to W?⍭ Generate me a similar question◊

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