MATH 40 LECTURE 7: INVERTIBLE MATRICES
DAGAN KARP
In this lecture we dene what it means for a matrix to be invertible, discuss rst properties and examples of invertible matrices, determine criteria for invertibility, and see a
deep connection between the
MATH 40 LECTURE 9: SUBSPACE
DAGAN KARP
What does it mean for one vector space to live inside of another vector space? For
example, isnt every line a copy of R1 ? So, isnt any line in R3 an example of one vector
space living inside of another? And how abou
MATH 40 LECTURE 10: BASES, DIMENSION AND RANK
DAGAN KARP
In this lecture we return to our discussion of subspaces. We learn what a basis is, and
use it to dene the dimension of a subspace. We also revisit the notion of rank, and obtain
a second part of th
MATH 40 LECTURE 4: GAUSSIAN ELIMINATION
DAGAN KARP
In our last lecture, we were introduced to the notion that matrices are useful tools for
solving linear systems in two variables. In this lecture, we extend this to higher dimensions, point out key ideas
MATH 40 LECTURE 3: INTRODUCTION TO SYSTEMS OF LINEAR
EQUATIONS
DAGAN KARP
What is the intersection of two lines in the real plane R2 ? In other words, how many
points are in the intersection of two lines in the plane? The answer, of course, is zero, or
on
MATH 40 LECTURE 6: LINEAR INDEPENDENCE CONTINUED
AND MATRIX OPERATIONS
DAGAN KARP
In this lecture we continue our study of linear independence, and then discuss basic
matrix operations.
Recall, we ended last time with the following theorem, which characte
MATH 40 LECTURE 11: LINEAR TRANSFORMATIONS
DAGAN KARP
In this lecture, we provide one interpretation of a matrix, giving the notion of matrix
greater depth than just an array of numbers.
Denition 1. A linear transformation from Rn to Rm is a map T : Rn Rm
MATH 40 LECTURE 1: VECTORS AND THE DOT PRODUCT
DAGAN KARP
Our goal in this course is to begin a study of the beautiful world of the linear linear
objects, linear operators, their algebra and even their geometry. Said differently, we hope
to study some of
MATH 40 LECTURE 2: PROJECTIONS AND PLANES
DAGAN KARP
In our last lecture we explored the dot product. We begin with a continuation of this
exploration.
1. D OT P RODUCT AND P ROJECTIONS
Theorem 1. For any vectors u and v, we have
u v = |u| |v| cos ,
where
MATH 40 LECTURE 8: THE FUNDAMENTAL THEOREM OF
INVERTIBLE MATRICES
DAGAN KARP
In our last lecture we were introduced to the notion of the inverse of a matrix, we used
the Gauss-Jordan method to nd the inverse of a matrix, and we saw that any linear system
MATH 40 LECTURE 5: LINEAR INDEPENDENCE AND SPAN
DAGAN KARP
In this lecture we continue our study of linear systems. In particular, we develop further techniques in our use of matrices to solve linear systems. Along the way, we encounter important notions