### Linear Algebra and Geometry. Exam feb. 18, 2015.

### 1. Let u = (1, 1, 1, −1), v = (1, 1, 1, 1), w = (1, 0, 0, 0).

### (a) Find v ^{0} and w ^{0} ∈ V _{4} such that: v ^{0} is a linear combination of u and v, w ^{0} is a linear combination of u, v and w, and {u, v ^{0} , w ^{0} } is an orthogonal set.

### (b) Write (0, 1, 0, 0) as the sum of a linear combination of u , v, w and of a vector orthogonal to u, v, w.

### Hint

### (a) The request can be summarized as: find v ^{0} ∈ L(u, v) and w ^{0} ∈ L(u, v, w) such that {u, v ^{0} , w ^{0} } is an orthogonal set. Therefore what is required is just the usual Gram=Schmidt orthogonalization of the vectors u, v, w.

### (b) By the Orthogonal Decomposition Theorem there is a unique solution to this problem:

### (0, 1, 0, 0) = pr _{L(u,v,w)} ((0, 1, 0, 0)) + pr _{L(u,v,w)}

^{⊥}

### ((0, 1, 0, 0))

### To find the two projections, one first observes that it fact it is enough to compute one one of them: again by the Orthogonal Decomposition Theorem one projection determines the other one (for example: pr _{L(u,v,w)} ((0, 1, 0, 0)) = (0, 1, 0, 0) − pr _{L(u,v,w)}

^{⊥}

### ((0, 1, 0, 0)) ).

### To compute one projection one can proceed in two different ways:

### (i) Compute pr _{L(u,v,w)} ((0, 1, 0, 0)). For this one uses point (a): since {u, v ^{0} , w ^{0} } is an orthogonal basis of L(u, v, w) then

### pr _{L(u,v,w)} ((0, 1, 0, 0)) = pr _{L(u)} ((0, 1, 0, 0)) + pr _{L(v}

^{0}

_{)} ((0, 1, 0, 0)) + pr _{L(w)}

^{0}

### ((0, 1, 0, 0)) =

### = (0, 1, 0, 0) · u

### (u, u) u + (0, 1, 0, 0) · v ^{0}

### (v ^{0} , v ^{0} ) v ^{0} + (0, 1, 0, 0) · w ^{0} (w ^{0} , w ^{0} ) w ^{0}

### (ii) Compute pr _{L(u,v,w)}

^{⊥}

### ((0, 1, 0, 0)). For this one first computes a basis of L(u, v, w) ^{⊥} (which is one-dimensional). For this one just needs to solve the system (a very easy one!)

###

###

###

### (x, y, z, t) · u =0 (x, y, z, t) · v =0 (x, y, z, t) · w =0

### . The space of solutions is L(u, v, w) ^{⊥} and it will be of the form L(h). Then

### pr _{L(u,v,w)}

^{⊥}