The kernel of a matrix transformation TA: Rn →Rm is a subspace of Rm.
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Yes, the kernel of a matrix transformation TA: Rn →Rm is a subspace of Rm.
The kernel of a matrix transformation is also known as the null space. It is the set of all vectors in the domain (Rn) that are mapped to zero in the codomain (Rm) by the given matrix transformation.
To prove that the kernel is a subspace of Rm, we need to show that it satisfies the three properties of a subspace:
1. Closed under addition: If x and y are in the kernel of TA, then TA(x + y) = TA(x) + TA(y) = 0 + 0 = 0. Therefore, x + y is also in the kernel of TA.
2. Closed under scalar multiplication: If x is in the kernel of TA, then for any scalar c, TA(cx) = cTA(x) = c(0) = 0. Therefore, cx is also in the kernel of TA.
3. Contains the zero vector: Since TA(0) = 0 for any matrix transformation, the zero vector is always in the kernel of TA.
Thus, the kernel of a matrix transformation TA: Rn →Rm satisfies all the properties of a subspace, and is therefore a subspace of Rm.
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