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Question 12 (6 marks) Consider the inner product on $mathbb{R}^2$ defined by $langle (u_1, u_2), (v_1, v_2) angle = u_1v_1 + u_2v_2 + (u_1 + u_2)(v_1 + v_2)$. (a) Show that this inner product can be written in the form $langle (u_1, u_2), (v_1, v_2) angle = egin{bmatrix} u_1 & u_2 end{bmatrix} egin{bmatrix} a & b \ c & d end{bmatrix} egin{bmatrix} v_1 \ v_2 end{bmatrix}$ for certain $a, b, c, d in mathbb{R}$. (b) Verify that axiom 4 for an inner product is satisfied. (c) Using this inner product, find a unit vector orthogonal to the vector $(2, 1)$.

          Question 12 (6 marks)
Consider the inner product on $mathbb{R}^2$ defined by
$langle (u_1, u_2), (v_1, v_2) 
angle = u_1v_1 + u_2v_2 + (u_1 + u_2)(v_1 + v_2)$.
(a) Show that this inner product can be written in the form
$langle (u_1, u_2), (v_1, v_2) 
angle = egin{bmatrix} u_1 & u_2 end{bmatrix} egin{bmatrix} a & b \ c & d end{bmatrix} egin{bmatrix} v_1 \ v_2 end{bmatrix}$
for certain $a, b, c, d in mathbb{R}$.
(b) Verify that axiom 4 for an inner product is satisfied.
(c) Using this inner product, find a unit vector orthogonal to the vector $(2, 1)$.

Question 12 (6 marks)
Consider the inner product on mathbbR^2 defined by
langle (u1, u2), (v1, v2)
angle = u1v1 + u2v2 + (u1 + u2)(v1 + v2).
(a) Show that this inner product can be written in the form
langle (u1, u2), (v1, v2)
angle = eginbmatrix u1 u2 endbmatrix eginbmatrix a b c d endbmatrix eginbmatrix v1 v2 endbmatrix
for certain a, b, c, d in mathbbR.
(b) Verify that axiom 4 for an inner product is satisfied.
(c) Using this inner product, find a unit vector orthogonal to the vector (2, 1).

Added by Benjamin M.

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