CH1p1.pdf - Math 4242 Applied Linear Algebra Introduction...

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Math 4242 - Applied Linear Algebra Introduction to Linear Systems these notes follow closely Olver and Shakiban 1.1-1.4 Natalie E. Sheils 1 Why study linear algebra? In applications, the size of systems can be huge! In image processing or mathematical biology, for instance, there can be millions of equations involving millions of unknowns. Linear algebra is the systematic, mathematical method of solving these large systems. Example 1 Suppose we observe an object that travels in the shape of a parabola modeled by a quadratic polynomial p ( t ) . We know the velocity at t = 2 is 0, the position at t = 6 is 12, and the position at t = 0 is 0. That is, p ( t ) = at 2 + bt + c and 4 a + b + 0 = 0 36 a + 6 b + c = 12 0 + 0 + c = 0 . This problem is linear because the unknowns appear to the first power. That is, there are no terms like b 2 or ab . One way to solve this system is reducing it to triangular form by adding and subtracting multiples of one equation to another. We begin by multiplying the first equation by - 9 and adding it to the second equation. Then we have 4 a + b + 0 = 0 0 - 3 b + c = 12 0 + 0 + c = 0 . Now, we can use back substitution to solve this system. The third equation tells us c = 0 . Then, subbing that into the second equation, we see that - 3 b = 12 b = - 4 . With this information we can solve the first equation for a . The solution to this problem is p ( t ) = t 2 - 4 t as shown in Figure 1. 2 Matrices and Vectors Lets define some matrices to use later: A = 1 2 - 1 0 , (1) 1
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1 2 3 4 5 6 t 5 10 p ( t ) Figure 1: The function p ( t ) which is constructed by giving data at three points. and B = 3 - 5 2 1 , (2) C = 1 - 5 2 2 3 5 . (3) A and B are 2 × 2 while C = 2 × 3. Note that we always say the size of a matrix is the number of rows by the number of columns. 2.1 Matrix Operations You can add and subtract matrices element by element, however, the matrices must be the same size. For example, A + B = 1 2 - 1 0 + 3 - 5 2 1 = 4 - 3 1 1 . We can also perform scalar multiplication - 2 A = - 2 - 4 2 0 . Matrix multiplication is not done element-wise. Instead we multiply the rows of the first matrix by the columns of the second matrix. AB = 1 2 - 1 0 3 - 5 2 1 = 7 - 3 - 3 5 . 2
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We can multiply matrices of different sizes as long as the number of columns in first matrix is the same as the number of rows in the second matrix. That is, one can find the product BC but the product CB does not exist. Also, B is 2 × 2 and C is 2 × 3 so BC is 2 × 3. We must define a few terms. The zero matrix is a matrix whose entries are all 0. This matrix is the additive identity element. The identity matrix is a square matrix with 1s on the main diagonal and zeros elsewhere. This matrix is the multiplicative identity element. We denote this n × n matrix as I or I n . For example, I 2 B = 1 0 0 1 3 - 5 2 1 = 3 - 5 2 1 = BI 2 . The identity matrix is an example of a diagonal matrix and it commutes with all other matrices.
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