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By Bacco M., Mocellin V.

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12. (a) The product formula holds if A is an elementary matrix; this is a consequence of the determinant axioms coupled with the fact that elementary matrices are obtained by applying the corresponding row operation to the identity matrix, with det I = 1. 82) also holds. 25 then implies that the product formula is valid whenever A is nonsingular. 51 to Z and Z B. 20, every singular matrix can be written as A = E1 E2 · · · EN Z, where the Ei are elementary matrices, while Z, its row echelon form, is a matrix with a row of zeros.

Cases (b), (c), (e) span R 2 . 5. 6. They are the same. Indeed, since v1 = u1 + 2 u2 , v2 = u1 + u2 , every vector v ∈ V can be written as a linear combination v = c1 v1 + c2 v2 = (c1 + c2 ) u1 + (2 c1 + c2 ) u2 and hence belongs to U . Conversely, since u1 = − v1 + 2 v2 , u2 = v1 − v2 , every vector u ∈ U can be written as a linear combination u = c1 u1 + c2 u2 = (− c1 + c2 ) v1 + (2 c1 − c2 ) v2 , and hence belongs to U . 7. (a) Every symmetric matrix has the form 49 a b b c ! =a 1 0 0 0 ! +c 0 0 0 1 !

X (b) For instance, the zero vector 0 = ( 0, 0, 0 )T does not satisfy the equation. 2. (b,c,d,g,i) are subspaces; the rest are not. Case (j) consists of the 3 coordinate axes and the line x = y = z. 3. 4. Any vector of the form 2C 2a − c C A + b @ 0 A + c @ −1 A = @ A = @ y A will −1 1 3 −a + b + 3c z 1 0 1 2 0 C belong to W . The coefficient matrix B @ 2 0 −1 A is nonsingular, and so for any −1 1 3 aB @ 45 x = ( x, y, z )T ∈ R 3 we can arrange suitable values of a, b, c by solving the linear system.

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