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Geometry of row operations and determinants

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Why matrix multiplication works the way it does

Matrix theorem

If you are finished early then go to the computer lab and work on the final project [I will bring references to help look up teaching ideas].

Finish the last slide of transformations. Computer graphics demo via definition of triangle := Matrix([[4,4,6,4],[3,9,3,3],[1,1,1,1]]); and then ASULearn Computer Graphics Example D. Also look at Homogeneous 3D coordinates and Example G. Example I.

transformations *Note the typo in the rotation matrix

Prove that a rotation matrix rotates algebraically as well as geometrically. Discuss dilation, shear, and reflection. Discuss what Euclidean transformation is missing from our list.

Clicker questions

Clicker questions

Prove that lambda=0 iff A is not invertible.

Prove that x is and eigenvector iff cx is an eigenvector for c nonzero.

Prove that if there are n distinct eigenvalues then the matrix is diagonalizable.

Give a counterexample to the reverse implication. Go to the computer lab:

Group work

If you are done before we come back together, then work on homework for tomorrow [Schaum's outline chapters 5 and 6]

Revisit linear transformations and prove that differentiation is linear.

Coffee mixing problem and numerical methods issue related to decimals versus fractions. Algebra and geometry of linear combinations of vectors. Basis for the row space and column space.

Define eigenvalues and eigenvectors [Ax=lambdax, vectors that are scaled on the same line through the origin, matrix multiplication is turned into scalar multiplication]. Prove that the computational methods work and explain why we need the extra equation given by the characteristic polynomial. Prove that high powers are easy to create. Explore the eigenvector decomposition formula:

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Dynamical systems demo.

Diagonalizability.

Applications: Traffic and oil pipeline construction. Spanning trees.

group work

Linear Transformations

Prove that any orthogonal set of n vectors in R

Vector spaces and bases associated to a coefficient matrix: Row space, Column space = Range. Rank. Prove that the dimension row space= dimension of the column space.

Vector space associated to homogeneous solutions: the nullspace. Nullity and relationship to rank.

span and li group work clicker review

Prove that if cv=0 then c=0 or v=0.

Is the Range or Image of a matrix subspaces of R

Prove that if a matrix equation has 2 distinct solutions then it has infinite solutions.

Representations of spaces. Graph paper, span, li, basis and dimension.

a) yes

b) no

Generating subspaces of R

Are the following subspaces? Rotation matrices, det 0 matrices, det non-zero matrices, stochastic matrices, continuous functions, R

The vector space axioms allow us to have a notion of flatness or linearity even for abstract items.

Continue Scalars, vectors and vectors spaces. Prove that the norm of of (u+v) = the norm of u plus the norm of v iff they are in the same direction. Look at subsets of R

Scalars, vectors and associativity. Exponentiation. Affine transformations as an operation on R

Introduction to Linear Algebra reading. Yarn activity.