In this assignment you will use concepts from linear algebra to solve several problems in the application domain of communication.

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You observe two encoded messages as they are being transmitted:

Determine which two unencoded vectors Alice sent, and then use them along with the encoded vectors and some matrix
operations to provide a definition of the encoding matrix. You should be able to use \augment
to accomplish this using a definition
that fits on a single line.
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Alice and Bob will agree on some u, M, and M' before parting. When sending a vector with x and y components, Alice will use the device to transmit two vectors: (x ⋅ u) and (y ⋅ u). Bob will receive w and w' as defined below, and will decode v by computing M w + M w'. Find u,M, and M' that make it possible to retrieve v, and complete the argument below. Hint: review the propositions available in the verification system that deal with matrices and vectors, as they may allow some algebraic manipulations to be more concise.
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We do not know s and r, but we do know all of the following:

One way to remove the [b;d] term from [x;y] is to first set up a matrix equation and solve for s and r:
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However, suppose we instead want to find a linear transformation f such that for any [x;y] as defined above, we have:
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In fact, given the information we have, we can construct exactly such a linear transformation:
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Complete the argument below showing that this linear transformation indeed has this property.




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Suppose you sample the radio signals at a given moment and obtain the vectors P below. Determine which scalars Alice, Bob, and Carol are each transmitting. Your answer can be in the form of a vector.
