Gaussian Elimination Quiz Demonstrate that matrices’ multiplication are not commutative (error: didn’t consider m\times m) Which 2\times 2 matrices under multiplication form a group? (error: closure need to proved on invertable matrices under multiplication, not just 2\times 2) Deriving Rotation matrices (error: clockwise vs counter-clockwise) Linear Independence Quiz Connection between linear independence and systems equations (error: beated around the bush) — the matrix of an nxn system of equations has a solution if the matrix’s column vectors is linearly independent Basis and Dimension Quiz put 0 into a basis AAAA not lin. indep; figure out what the basis for a polynomial with a certain root is: it is probably of dimension m (instead of m+1), because scalars doesn’t work in the case of p(3)=0; so basis is just the scalars missing some inequality about basis? — its just that lin.idp sets is shorter or equal to basis and spanning sets is longer or equal to basis Final, part 1 definition of vector space: scalar multiplication is not an operation straight forgot dim(U+V) = dim U + dim V - dim (U\cap V) plane containing (1,0,2) and (3,-1,1): math mistake proof: det A det B = det AB Final, part 2 Counterproof: If v_1 \dots v_4 is a basis of V, and U is a subspace of V with v_1, v_2 \in U and v_3, v_4 not in U, v_1, v_2 is a basis of U Counterproof: if T \in \mathcal{L}(V,V) and T^{2}=0, then T=0 Counterproof: if s,t \in \mathcal{L}(V,V), and ST=0, then null\ s is contained in range\ T Product Spaces Quiz Prove that \mathcal{L}(V_1 \times V_2 \times \dots \times V_{m}, W) and \mathcal{L}(V_1, W) \times \dots \times \mathcal{L}(V_{m}, W) are isomorphic error: didn’t do it Quotient Spaces Quiz Couldn’t prove that the list in linearly independent: the linear combinations is some c_1v_1 + \dots c_{m}v_{m} + U; as v_1 \dots v_{m} is a basis of V / U, c_1 \dots c_{m} = 0, now the second part is also a basis so they are 0 too. The spanning proof: v + U = , rewrite as basis, etc.