16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 2.3.7

Vector Spaces

Mathematics — Linear Algebra · subspaces-of-matrix-space-dimensions

Problem

Let VV be the vector space of Exercise 6. Let W1W_1 be the set of matrices of the form [x−xyz]\begin{bmatrix} x & -x \\ y & z \end{bmatrix} and let W2W_2 be the set of matrices of the form [ab−ac].\begin{bmatrix} a & b \\ -a & c \end{bmatrix}. (a) Prove that W1W_1 and W2W_2 are subspaces of VV. (b) Find the dimensions of W1W_1, W2W_2, W1+W2W_1 + W_2, and W1∩W2W_1 \cap W_2.
Intuitive explanation

A subspace is closed under addition and scalar multiplication, and contains the zero element—check these three properties for each set by using the constraint structure. For dimension, count the free parameters in each subspace's general form, then use the dimension formula dim⁡(W1+W2)=dim⁡(W1)+dim⁡(W2)−dim⁡(W1∩W2)\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2) to find the intersection's dimension without computing it directly.

Intuitive explanation

A subspace is closed under addition and scalar multiplication, and contains the zero element—check these three properties for each set by using the constraint structure. For dimension, count the free parameters in each subspace's general form, then use the dimension formula dim⁡(W1+W2)=dim⁡(W1)+dim⁡(W2)−dim⁡(W1∩W2)\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2) to find the intersection's dimension without computing it directly.

Related theory

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