UNH Linear Algebra Linear Systems Projections Projection Error & Matrixes Worksheet


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Given three points (ti,bi) =(-1,7),(1,7),(2,21) and a linear model b=C+Dr:
1. (1 point) Write the three equations for the line to go through the three points and give the corresponding linear
system Ax=b. (i.e. give A, x, and b). Hint: it does not have a solution
2. (1 point) Give and solve the normal equations in Ź to find the projection p=Aſ of b onto the column space of
A, instead (1. e. find & with least ||AR-b|?).
3. (1 point) Compute the error between b and its projection p= A.
4. (1 point) Consider a 5 x 4 matrix A, with rank 2. Complete the following:
The column space, C(A), is a subspace of R__and has dimension — Its orthogonal complement is the
space, and has dimension — The row space, C(AT), is a subspace of RA
and has dimension Its orthogonal complement is the
and has di-
5. (1 point) Let V = span
– (14)
be a one-dimensional subspace of R3.
We want to find its orthogonal complement V+. Using orthogonality of the four fundamental subspaces and
how we find a basis for the null space, find a basis for V. Hint: it is easy to construct a matrix with prescribed
row space C(AT).

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