If a is not equal to c, then the equation has exactly one solution. Since a does not equal c, the equation has exactly one solution. Choose one of the possibilities from question 1 and explain why this occurs under the required restrictions on a, b, c, and d.
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In each case, we can use inverse operations to illustrate. When a does not equal c, we use inverse operations to solve the linear equation: We see that this is the only value of x that satisfies the equation, so the equation has only one solution. Now we subtract ax from both sides.
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We see that: This is a false statement since b does not equal d. And yes, when we decide to treat inputs as vector coordinates, the operations matrix will transform our vectors.
These are geometric interpretations of multiplication, and how to warp a vector space. Just remember that vectors are examples of data to modify. These dollar values could come from another matrix that multiplied the number of shares by their price. Fancy that! We want to adjust each stock value, using something similar to the identity matrix:.
Making sense? Three inputs enter, four outputs leave. We can crunch the numbers by hand, or use a Wolfram Alpha calculation :. The Transpose operation, indicated by t tau , converts rows to columns. Linear algebra emerged in the s yet spreadsheets were invented in the s. I blame the gap on poor linear algebra education. As the matrix evolves into the identity matrix, the values of x, y and z are revealed on the output side. This process, called Gauss-Jordan elimination, saves time. Words have technical categories to describe their use nouns, verbs, adjectives.
Matrices can be similarly subdivided. If the input was a unit vector representing area or volume of 1 , the determinant is the size of the transformed area or volume. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix. What if we want the metric version? Take the instructions, treat them like text, and convert the units. An operations matrix is similar: commands to modify.
Applying one operations matrix to another gives a new operations matrix that applies both transformations, in order. The beauty of linear algebra is representing an entire spreadsheet calculation with a single letter. Want to apply the same transformation a few times? Yes, because you asked nicely. Time to expand our brains.
The Rank of a Matrix
Now our operations matrix has an extra, known value to play with! Want the geeky explanation? We skew that higher dimension, which looks like a slide in the current one. For example: take input x, y, z, 1 and run it through:. We keep the dummy entry, and can do more slides later. BetterExplained helps k monthly readers with friendly, insightful math lessons more.
An Intuitive Guide to Linear Algebra. We missed the key insight: Linear algebra gives you mini-spreadsheets for your math equations. Organizing Inputs and Operations Most courses hit you in the face with the details of a matrix. Grammar is not the focus. First, how should we track a bunch of inputs?
How about a list: x y z Not bad. We could write it x, y, z too — hang onto that thought.
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Only need the first input? We should separate the inputs into groups: 1st Input 2nd Input a x b y c z Much neater. And how could we run the same input through several operations? Have a row for each operation: F: 3 4 5 G: 3 0 0 Neat. Time for the red pill.
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