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Check list for Midterm II

- Linear spaces and
subspaces:
- What is a subspace of a
vector space?

- How do you check a subset
of a linear space is a subspace?
- What is the dimension of a
linear space?

- Find a basis of a given
linear space and determine its
dimension.
- Which vector spaces are
called finite dimensional?

- Important examples of
vector spaces: the coordinate space R
^{n},
the space M_{nm}
of all nxm matrices, the space P_{n}
of all polynomials of degree less or equal n.

- Linear transformations
- Determine when a
transformation is linear.

- Find the matrix of a
linear transformation with respect to a
given basis.

- Find the kernel, image,
rank and nullity of a linear
transformation between linear spaces.
- Find the basis of the
kernel and image of a linear
transformation.

- Determine whether a linear
transformation is an isomorphism.
- What is a relation between
matrices of a linear map with
respect to two different bases?

- Coordinates
- Find the change of
coordinates matrix between two given basis
- Find the coordinates of a
vector with respect to a basis

- Orthogonality
- Determine whether a basis is orthonormal.
- Find the orthogonal projection of a vector in a linear space V
onto a subspace of V.
- Find the orhogonal complement of a subspace of a linear space.
- Compute the length of a vector.
- Find the angle between two vectors.
- Perform the Gram-Schmidt process
- Find the QR factorization of a matrix.

As usual, going over the past homeworks provides the best preparation.
For extra practice, the following exercises are useful:

3.4: 13, 21, 27, 39;

4.1: 2,7,10, 25, 27, 31;

4.2: 7, 23, 25, 53, 67;

4.3: 7, 21, 24,42;

5.1: 7, 15, 21, 26, 29

Topics you need to know (this is list includes just the
highlights)

- What is a linear combination
of vectors?
- What is a span of vectors?
- Which vectors are said to be
linearly dependent?
- Which vectors are said to be
linearly independent?
- The dimension is the number
of vectors in a basis.
- The dimension is the maximal
number of linearly independent
vectors.
- The dimension is the minimal
number of spanning vectors.
- What is a linear
transformation?
- What is the kernel of a
linear transformation?
- What is the image of a
linear transformation?
- The rank of a linear
transformation is the dimension of the
image.
- What does the Kernel-Image
(Rank-Nullity) theorem say?
- What is an isomorphism?
- Which vectors are said to be
orthogonal?