commit 8fe300af4e072eb58af851927f2c950378141792
parent 0208950102328e23435e869e2f7141749d20acc1
Author: Andrew <andrewlaack1@gmail.com>
Date: Wed, 24 Jul 2024 14:06:27 -0500
Took notes on lin alg for the day
Diffstat:
4 files changed, 21 insertions(+), 0 deletions(-)
diff --git a/Image.md b/Image.md
@@ -18,3 +18,5 @@ The result of the tranformation of a subspace is the image of the subspace under
Ex.
T(V) = image of V under T
+
+We call this the image of T stated as im(T) when we are referring to any vector in R^n not necessarily a given subspace. The distinction here is that T(V) defines V as the codomain whereas im(T) defines the codomain as all possible vectors in R^n.
diff --git a/JointDensityFunction.md b/JointDensityFunction.md
@@ -0,0 +1,10 @@
+:prob:
+# Joint Density Function
+
+Prob L9
+
+## Notes
+
+**Definition:** A joint density function is a function that takes two inputs and outputs a probability of the combination.
+
+We can define the function as f_{XY} : R^2 -> R such that for all A in R^2 we have P((X,Y) in A) = integral(integral(f_XY(x,y)))
diff --git a/LinearTransformation.md b/LinearTransformation.md
@@ -40,3 +40,9 @@ When describing LTs in matrix form each column represents where a given unit vec
**Important:**
Any LT can be represented as a matrix and all matrix multiplication is a LT.
+
+## Image
+
+The image of a linear transformation (im(T)) are all possible outputs of the function where the inupts of T are any vector in R^n.
+
+The image of Z under T are all possible outputs of the function with inputs that are in Z.
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -78,3 +78,6 @@ L8:
- [[MixedRandomVariable.md]]
- [[NormalDistribution.md]]
- [[BernoulliRandomVariable.md]]
+L9:
+ - [[JointDensityFunction.md]]
+ - [[]]
