I have several subspaces where I have to determine their dimension and whether they are affine or linear? These are my answers- are they correct? Thanks for help!
a) $X = \{ x \in R^n | a^Tx = 0 \}, a \in R^n$ is given
- linear since any linear combination $\alpha x + \beta y, x, y \in X$ is in X. I believe its affine, too, since actual coefficients $\alpha, \beta$ doesnt really matter in this case because $\alpha(a_1x_1 + \dots + a_nx_n) + \beta(a_1y_1 + a_ny_n) = 0$ every time...
- dimension? Im guessing its max $n-1$ but Im not sure
b) $X = \{ x \in R^n | a^Tx = c \}, a \in R^n, c \in R$ are given
- affine because in order for the linear combination to be in $X$, $\alpha_i$ must sum to 1
- dimension again max $n-1$?
c) $X = \{ x \in R^n | x^Tx = 1 \}$
- affine (same justification as in b))
d) $X = \{ x \in R^n | a^Tx = I \}, a \in R^n$ is given
- not sure at all Are my assumptions correct